Moment estimates for Lévy Processes

Abstract

For real Lévy processes $(X\_t)\_{t \geq 0}$ having no Brownian component with Blumenthal-Getoor index $β$, the estimate $\E \sup\_{s \leq t} | X\_s - a\_p s |^p \leq C\_p t$ for every $t \in [0,1]$ and suitable $a\_p \in \R$ has been established by Millar \cite{MILL} for $β< p \leq 2$ provided $X\_1 \in L^p$. We derive extensions of these estimates to the cases $p > 2$ and $p \leqβ$.